- Open Access
Direct evidence of a near-ideal ground state in triangular-lattice
Phys. Rev. Materials 10, 025004 – Published 26 February, 2026
DOI: https://doi.org/10.1103/65bn-63kj
Abstract
We investigated the local Co electronic structure of using polarization-dependent x-ray absorption spectroscopy (XAS) in combination with full multiplet cluster calculations. We employed the line-fitting inverse partial fluorescence yield (IPFY) technique to obtain accurate XAS spectra from strong insulating materials. Our combined experimental and theoretical analysis reveals a very small effective trigonal distortion of only 11 meV in the octahedra, indicating a close to ideal condition to render a ground state with the character. With our cluster model we were also able to simulate magnetic susceptibility measurements along different directions in the crystal. These findings highlight as a promising platform for exploring exotic magnetic phenomena associated with ground states on triangular lattices.
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References (54)
- P. Anderson, Resonating valence bonds: A new kind of insulator? Mater. Res. Bull. 8, 153 (1973).
- A. V. Chubukov and D. I. Golosov, Quantum theory of an antiferromagnet on a triangular lattice in a magnetic field, J. Phys.: Condens. Matter 3, 69 (1991).
- O. A. Starykh, Unusual ordered phases of highly frustrated magnets: A review, Rep. Prog. Phys. 78, 052502 (2015).
- B. Bernu, C. Lhuillier, and L. Pierre, Signature of Néel order in exact spectra of quantum antiferromagnets on finite lattices, Phys. Rev. Lett. 69, 2590 (1992).
- B. Bernu, P. Lecheminant, C. Lhuillier, and L. Pierre, Néel order versus spin liquid in quantum Heisenberg antiferromagnets on triangular and Kagomé lattices, Phys. Scr. 1993, 192 (1993).
- B. Bernu, P. Lecheminant, C. Lhuillier, and L. Pierre, Exact spectra, spin susceptibilities, and order parameter of the quantum Heisenberg antiferromagnet on the triangular lattice, Phys. Rev. B 50, 10048 (1994).
- S. R. White and A. L. Chernyshev, Neèl order in square and triangular lattice Heisenberg models, Phys. Rev. Lett. 99, 127004 (2007).
- D.-V. Bauer and J. O. Fjærestad, Schwinger-boson mean-field study of the Heisenberg quantum antiferromagnet on the triangular lattice, Phys. Rev. B 96, 165141 (2017).
- F. Ferrari and F. Becca, Dynamical structure factor of the Heisenberg model on the triangular lattice: Magnons, spinons, and gauge fields, Phys. Rev. X 9, 031026 (2019).
- S.-S. Gong, W. Zhu, J.-X. Zhu, D. N. Sheng, and K. Yang, Global phase diagram and quantum spin liquids in a spin- triangular antiferromagnet, Phys. Rev. B 96, 075116 (2017).
- S. Hu, W. Zhu, S. Eggert, and Y.-C. He, Dirac spin liquid on the spin- triangular Heisenberg antiferromagnet, Phys. Rev. Lett. 123, 207203 (2019).
- Y. Iqbal, W.-J. Hu, R. Thomale, D. Poilblanc, and F. Becca, Spin liquid nature in the Heisenberg triangular antiferromagnet, Phys. Rev. B 93, 144411 (2016).
- P. A. Maksimov, Z. Zhu, S. R. White, and A. L. Chernyshev, Anisotropic-exchange magnets on a triangular lattice: Spin waves, accidental degeneracies, and dual spin liquids, Phys. Rev. X 9, 021017 (2019).
- S. N. Saadatmand and I. P. McCulloch, Detection and characterization of symmetry-broken long-range orders in the spin- triangular Heisenberg model, Phys. Rev. B 96, 075117 (2017).
- Z. Zhu and S. R. White, Spin liquid phase of the Heisenberg model on the triangular lattice, Phys. Rev. B 92, 041105(R) (2015).
- Z. Zhu, P. A. Maksimov, S. R. White, and A. L. Chernyshev, Topography of spin liquids on a triangular lattice, Phys. Rev. Lett. 120, 207203 (2018).
- Y. Li, P. Gegenwart, and A. A. Tsirlin, Spin liquids in geometrically perfect triangular antiferromagnets, J. Phys.: Condens. Matter 32, 224004 (2020).
- R. Zhong, S. Guo, G. Xu, Z. Xu, and R. J. Cava, Strong quantum fluctuations in a quantum spin liquid candidate with a Co-based triangular lattice, Proc. Natl. Acad. Sci. USA 116, 14505 (2019).
- N. Li, Q. Huang, X. Y. Yue, W. J. Chu, Q. Chen, E. S. Choi, X. Zhao, H. D. Zhou, and X. F. Sun, Possible itinerant excitations and quantum spin state transitions in the effective spin-1/2 triangular-lattice antiferromagnet (), Nat. Commun. 11, 4216 (2020).
- S. Lee, C. H. Lee, A. Berlie, A. D. Hillier, D. T. Adroja, R. Zhong, R. J. Cava, Z. H. Jang, and K.-Y. Choi, Temporal and field evolution of spin excitations in the disorder-free triangular antiferromagnet , Phys. Rev. B 103, 024413 (2021).
- Y. Y. Huang, D. Z. Dai, C. C. Zhao, J. M. Ni, L. S. Wang, B. L. Pan, B. Gao, P. Dai, and S. Y. Li, Thermal conductivity of triangular-lattice antiferromagnet (): Absence of itinerant fermionic excitations, arXiv:2206.08866.
- L. Woodland, R. Okuma, J. R. Stewart, C. Balz, and R. Coldea, From continuum excitations to sharp magnons via transverse magnetic field in the spin- Ising-like triangular lattice antiferromagnet , Phys. Rev. B 112, 104413 (2025).
- J. Sheng, L. Wang, W. Jiang, H. Ge, N. Zhao, T. Li, M. Kofu, D. Yu, W. Zhu, J.-W. Mei, Z. Wang, and L. Wu, Continuum of spin excitations in an ordered magnet, Innovation 6, 100769 (2025).
- Y. Gao, Y.-C. Fan, H. Li, F. Yang, X.-T. Zeng, X.-L. Sheng, R. Zhong, Y. Qi, Y. Wan, and W. Li, Spin supersolidity in nearly ideal easy-axis triangular quantum antiferromagnet (), npj Quantum Mater. 7, 89 (2022).
- J. Xiang, C. Zhang, Y. Gao, W. Schmidt, K. Schmalzl, C.-W. Wang, B. Li, N. Xi, X.-Y. Liu, H. Jin, G. Li, J. Shen, Z. Chen, Y. Qi, Y. Wan, W. Jin, W. Li, P. Sun, and G. Su, Giant magnetocaloric effect in spin supersolid candidate (), Nature (London) 625, 270 (2024).
- T. Burnus, Z. Hu, H. H. Hsieh, V. L. J. Joly, P. A. Joy, M. W. Haverkort, H. Wu, A. Tanaka, H.-J. Lin, C. T. Chen, and L. H. Tjeng, Local electronic structure and magnetic properties of studied by x-ray absorption and magnetic circular dichroism spectroscopy, Phys. Rev. B 77, 125124 (2008).
- C. Wellm, W. Roscher, J. Zeisner, A. Alfonsov, R. Zhong, R. J. Cava, A. Savoyant, R. Hayn, J. van den Brink, B. Büchner, O. Janson, and V. Kataev, Frustration enhanced by Kitaev exchange in a triangular antiferromagnet, Phys. Rev. B 104, L100420 (2021).
- B. S. Mou, X. Zhang, L. Xiang, Y. Xu, R. Zhong, R. J. Cava, H. Zhou, Z. Jiang, D. Smirnov, N. Drichko, and S. M. Winter, Comparative Raman scattering study of crystal field excitations in Co-based quantum magnets, Phys. Rev. Mater. 8, 084408 (2024).
- G. Hussain, J. Zhang, M. Zhang, L. Yadav, Y. Ding, C. Zheng, S. Haravifard, and X. Wang, Experimental evidence of crystal-field, Zeeman-splitting, and spin-phonon excitations in the quantum supersolid (, Phys. Rev. B 111, 155129 (2025).
- T. I. Popescu, N. Gora, F. Demmel, Z. Xu, R. Zhong, T. J. Williams, R. J. Cava, G. Xu, and C. Stock, Zeeman split Kramers doublets in spin-supersolid candidate , Phys. Rev. Lett. 134, 136703 (2025).
- A. Tanaka and T. Jo, Resonant and photoemission in transition metal oxides predicted at threshold, J. Phys. Soc. Jpn. 63, 2788 (1994).
- F. M. F. de Groot, X-ray absorption and dichroism of transition metals and their compounds, J. Electron Spectrosc. Relat. Phenom. 67, 529 (1994).
- T. Burnus, Z. Hu, M. W. Haverkort, J. C. Cezar, D. Flahaut, V. Hardy, A. Maignan, N. B. Brookes, A. Tanaka, H. H. Hsieh, H.-J. Lin, C. T. Chen, and L. H. Tjeng, Valence, spin, and orbital state of Co ions in one-dimensional : An x-ray absorption and magnetic circular dichroism study, Phys. Rev. B 74, 245111 (2006).
- H.-J. Lin, Y. Y. Chin, Z. Hu, G. J. Shu, F. C. Chou, H. Ohta, K. Yoshimura, S. Hébert, A. Maignan, A. Tanaka, L. H. Tjeng, and C. T. Chen, Local orbital occupation and energy levels of Co in : A soft x-ray absorption study, Phys. Rev. B 81, 115138 (2010).
- Y. Y. Chin, Z. Hu, H.-J. Lin, S. Agrestini, J. Weinen, C. Martin, S. Hébert, A. Maignan, A. Tanaka, J. C. Cezar, N. B. Brookes, Y.-F. Liao, K.-D. Tsuei, C. T. Chen, D. I. Khomskii, and L. H. Tjeng, Spin-orbit coupling and crystal-field distortions for a low-spin state in , Phys. Rev. B 100, 205139 (2019).
- A. J. Achkar, T. Z. Regier, H. Wadati, Y.-J. Kim, H. Zhang, and D. G. Hawthorn, Bulk sensitive x-ray absorption spectroscopy free of self-absorption effects, Phys. Rev. B 83, 081106(R) (2011).
- A. J. Achkar, T. Z. Regier, E. J. Monkman, K. M. Shen, and D. G. Hawthorn, Determination of total x-ray absorption coefficient using non-resonant x-ray emission, Sci. Rep. 1, 182 (2011).
- H.-M. Tsai, H.-W. Fu, C.-Y. Kuo, L.-J. Huang, C.-S. Lee, C.-Y. Hua, K.-Y. Kao, H.-J. Lin, H.-S. Fung, S.-C. Chung, C.-F. Chang, A. Chainani, L. H. Tjeng, and C.-T. Chen, A submicron soft x-ray active grating monochromator beamline for ultra-high resolution angle-resolved photoemission spectroscopy, AIP Conf. Proc. 2054, 060047 (2019).
- M. M. Ferreira-Carvalho, S. Rößler, C. F. Chang, Z. Hu, S. M. Valvidares, P. Gargiani, M. W. Haverkort, P. K. Mukharjee, P. Gegenwart, A. A. Tsirlin, and L. H. Tjeng, Trigonal distortion in the Kitaev candidate honeycomb magnet (, Phys. Rev. B 112, 125135 (2025).
- M. W. Haverkort, M. Zwierzycki, and O. K. Andersen, Multiplet ligand-field theory using Wannier orbitals, Phys. Rev. B 85, 165113 (2012).
- M. W. Haverkort, G. Sangiovanni, P. Hansmann, A. Toschi, Y. Lu, and S. Macke, Bands, resonances, edge singularities and excitons in core level spectroscopy investigated within the dynamical mean-field theory, Europhys. Lett. 108, 57004 (2014).
- Y. Lu, M. Höppner, O. Gunnarsson, and M. W. Haverkort, Efficient real-frequency solver for dynamical mean-field theory, Phys. Rev. B 90, 085102 (2014).
- Quanty version 0.81, https://www.quanty.org.
- R. D. Cowan, The Theory of Atomic Structure and Spectra (University of California Press, Oakland, CA, 1981).
- A. E. Bocquet, T. Mizokawa, K. Morikawa, A. Fujimori, S. R. Barman, K. Maiti, D. D. Sarma, Y. Tokura, and M. Onoda, Electronic structure of early -transition-metal oxides by analysis of the core-level photoemission spectra, Phys. Rev. B 53, 1161 (1996).
- S. I. Csiszar, M. W. Haverkort, Z. Hu, A. Tanaka, H. H. Hsieh, H.-J. Lin, C. T. Chen, T. Hibma, and L. H. Tjeng, Controlling orbital moment and spin orientation in CoO layers by strain, Phys. Rev. Lett. 95, 187205 (2005).
- N. Hollmann, Z. Hu, T. Willers, L. Bohatý, P. Becker, A. Tanaka, H. H. Hsieh, H.-J. Lin, C. T. Chen, and L. H. Tjeng, Local symmetry and magnetic anisotropy in multiferroic and antiferromagnetic studied by soft x-ray absorption spectroscopy, Phys. Rev. B 82, 184429 (2010).
- K. Koepernik and H. Eschrig, Full-potential nonorthogonal local-orbital minimum-basis band-structure scheme, Phys. Rev. B 59, 1743 (1999).
- FPLO version 21.00, https://www.fplo.de.
- = 6.5 eV, = 8.2 eV, charge energy transfer = 6.5 eV, SOC = 0.066 eV, ionic crystal field = 0.395 eV, = 8 meV, hybridization = 2.45 eV, = 1.31 eV, = 1.29 eV (this hybridization values are then scaled to 83%), ligand crystal field = 0.693 eV. Slater integrals were reduced to 80% of the Hartree-Fock values.
- M. van Veenendaal, E. H. T. Poldi, L. S. I. Veiga, P. Bencok, G. Fabbris, R. Tartaglia, J. L. McChesney, J. W. Freeland, R. J. Hemley, H. Zheng, J. F. Mitchell, J.-Q. Yan, and D. Haskel, Electronic structure of Co states in the Kitaev material candidate honeycomb cobaltate probed with x-ray dichroism, Phys. Rev. B 107, 214443 (2023).
- G. A. Bain and J. F. Berry, Diamagnetic corrections and Pascal's constants, J. Chem. Educ. 85, 532 (2008).
- S. Agrestini, C.-Y. Kuo, K. Chen, Y. Utsumi, D. Mikhailova, A. Rogalev, F. Wilhelm, T. Förster, A. Matsumoto, T. Takayama, H. Takagi, M. W. Haverkort, Z. Hu, and L. H. Tjeng, Probing the ground state and the Van Vleck paramagnetism of the ions in layered , Phys. Rev. B 97, 214436 (2018).
- M. Carvalho, Direct evidence of a near-ideal = 1/2 ground state in triangular-lattice (2025), https://doi.org/10.17617/3.xe7bis.